lec4 - Introduction to Simulation - Lecture 4 Direct...

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Thanks to Deepak Ramaswamy, Michal Linear Systems Luca Daniel Introduction to Simulation - Lecture 4 Rewienski, Karen Veroy and Jacob White Direct Methods for Sparse 1
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Outline LU Factorization Reminder. Sparse Matrices – Struts and joints, resistor grids, 3-d heat flow Tridiagonal Matrix Factorization General Sparse Factorization – Fill-in and Reordering – Graph Based Approach Sparse Matrix Data Structures – Scattering 2
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22 23 24 31 32 33 34 42 43 44 M M M M M 44 M 43 M 43 M 44 M 34 M 33 M 32 M 42 M SMA-HPC ©2003 MIT 11 12 13 14 21 41 MM M M M M M M M M 43 33 M M 44 M LU Basics Picture Factoring 22 M 23 M 24 M 32 22 M M 42 M M 33 M 34 M 21 11 M M 31 11 M M 41 11 M M The above is an animation of LU factorization. In the first step, the first equation is used to eliminate x1 from the 2nd through 4th equation. This involves multiplying row 1 by a multiplier and then subtracting the scaled row 1 from each of the target rows. Since such an operation would zero out the a21, a31 and a41 entries, we can replace those zero’d entries with the scaling factors, also called the multipliers. For row 2, the scale factor is a21/a11 because if one multiplies row 1 by a21/a11 and then subtracts the result from row 2, the resulting a21 entry would be zero. Entries a22, a23 and a24 would also be modified during the subtraction and this is noted by changing the color of these matrix entries to blue. As row 1 is used to zero a31 and a41, a31 and a41 are replaced by multipliers. The remaining entries in rows 3 and 4 will be modified during this process, so they are recolored blue. This factorization process continues with row 2. Multipliers are generated so that row 2 can be used to eliminate x2 from rows 3 and 4, and these multipliers are stored in the zero’d locations. Note that as entries in rows 3 and 4 are modified during this process, they are converted to gr een. The final step is to used row 3 to eliminate x3 from row 4, modifying row 4’s entry, which is denoted by converting a44 to pink. It is interesting to note that as the multipliers are standing in for zero’d matrix entries, they are not modified during the factorization. 3
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LU Basics Factoring For i = 1 to n-1 { For j = i+1 to n { “For each target Row below the source” For k = i+1 to n { } } } Pivot ji ji ii M M M = Multiplier jk jk ji ik M M M M 2 1 1 ( ) 2 n i n ni = −= multipliers 1 2 3 1 2 ( ) 3 n i n = SMA-HPC ©2003 MIT Algorithm “For each Row” “For each Row element beyond Pivot” Multiply-adds 4
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LU Basics Dominant Matrices Factoring diagonally dominant matrix will never produce a zero pivot B) The matrix entries produced by LU factorization applied to a strictly diagonally dominant matrix will never increase by more 2 ) SMA-HPC ©2003 MIT Theorem about Diagonally A) LU factorization applied to a strictly than a factor (n-1 Theorem Gaussian Elimination applied to strictly diagonally dominant matrices will never produce a zero pivot.
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lec4 - Introduction to Simulation - Lecture 4 Direct...

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